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320                      ANALYTICAL MECHANICS
248. Damped Harmonic Motion. - When a par! icle moves in a harmonic field of force which is filled by a resist ing medium the motion of the particle is called damped harmonic motion. The particle is acted upon by two forces, namely, a harmonic force due to the field, and a resisting force due to the medium. All resisting forces are functions of the velocity and act in a direction opposed to that of the velocity. But since in harmonic motion the velocity does not attain great values, we can suppose the resist ing force to bo a linear function of the velocity. Therefore if /'' denotes the total force acting upon the particle we can write
where the first term of the right -hand member represents the harmonic force and the second term the resisting force. Substituting this value of F in the force equation we get
m* = -Av kr.                        (XIX)
A motion which is the perfect analogue of the motion defined by equation (XIX) is obtained when a rigid body placed in a resisting medium is subjected to a harmonic torque. The motion is defined by the following torque equation:
*f --*'*-*",                (XX)
where the first term of the right-hand member represents the harmonic torque and the second term the resisting torque. On account of the perfect analogy between the two types of motion a discussion of one of them is all that is necessary. We will consider the motion represented by equation (XX).
k"                    kf
Let  = 2 a and y = ft2, then equation (XX) becomes